Re: Permutations and Combinations

Two ladies must be in the committee.If two ladies are chosen, the number of combinations is 3C2.The remaining 4 members can be chosen from the 7 men and the number of combinations is 7C4.Hence, the number of ways in which two ladies can form part of the committee is 3C2*7C4.If three ladies are chosen, the number of combinations is 3C3.The remaining 3 members can be chosen from the remaining 7 men and the number of combinations is 7C3.Hence, the number of ways in which three ladies can form part of the committee is 3C3*7C3.The total number of combinations is 3C2*7C4 + 3C3*7C3.That is, 105 + 35 = 140.

Re: Permutations and Combinations

PC # 3

All of the numbers from 600 to 699 have a 6. This leaves 800 numbers between 100 and 999.Of those, 1/10 will have 6 as their tens digit. 800/10 = 80, so that leaves 720 more.Of those, 1/10 will have 6 as their units digit. 720/10 = 72, and adding this to 80 and 100 will give the answer.

Re: Permutations and Combinations

Re: Permutations and Combinations

There are 10 boxes numbered 1, 2, 3, 10. Each box is to be filled up either with a black or a white ball in such a way that at least 1 box contains a black ball and the boxes containing black balls are consecutively numbered. The total number of ways in which this can be done is..

Re: Permutations and Combinations

ganesh wrote:

Two ladies must be in the committee.If two ladies are chosen, the number of combinations is 3C2.The remaining 4 members can be chosen from the 7 men and the number of combinations is 7C4.Hence, the number of ways in which two ladies can form part of the committee is 3C2*7C4.If three ladies are chosen, the number of combinations is 3C3.The remaining 3 members can be chosen from the remaining 7 men and the number of combinations is 7C3.Hence, the number of ways in which three ladies can form part of the committee is 3C3*7C3.The total number of combinations is 3C2*7C4 + 3C3*7C3.That is, 105 + 35 = 140.

There are three ladies from which to choose the two slots designated for ladies. After those two slots are filled, there are eight unchosen members from which to randomly assign the remaining four committee seats. Why isn't the answer nCr(3,2)*nCr(8,4)=3*70=210? What am I missing? If I'm counting 70 possibilities twice, which possibilities am I double counting?

Edit to add: I'm reasonably confident the method I proposed is wrong, and Ganesh's solution is indeed correct. I'm interested in understanding why the method I proposed is wrong.

Re: Permutations and Combinations

Sudeep wrote:

There are 10 boxes numbered 1, 2, 3, 10. Each box is to be filled up either with a black or a white ball in such a way that at least 1 box contains a black ball and the boxes containing black balls are consecutively numbered. The total number of ways in which this can be done is..